Study of BMS-986205 and Nivolumab in Endometrial Cancer or Endometrial Carcinosarcoma That Has Not Responded to Treatment

Inclusion Criteria: * Subjects must have recurrent or persistent endometrial carcinoma (including: Endometrioid adenocarcinoma, serous adenocarcinoma, undifferentiated carcinoma, dedifferentiated carcinoma, clear cell adenocarcinoma, mixed epithelial carcinoma, adenocarcinoma not otherwise specified (N.O.S.), mucinous adenocarcinoma, squamous cell carcinoma, and transitional cell carcinoma) or endometrial carcinosarcoma). Histologic documentation of diagnosis of carcinoma is required. * All patients must have measurable disease. Measurable disease is defined by RECIST (version 1.1). Measurable disease is defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded). Each lesion must be ≥ 10 mm when measured by CT, MRI or caliper measurement by clinical exam; or ≥ 20 mm when measured by chest x-ray. Lymph nodes must be ≥ 15 mm in short axis when measured by CT or MRI. * Patients must have at least one "target lesion" to be used to assess response on this protocol as defined by RECIST version 1.1 (Section 12.0). Tumors within a previously irradiated field will be designated as "non-target" lesions unless progression is documented, or a biopsy is obtained to confirm persistence at least 90 days following completion of radiation therapy. * Females, age ≥ 18 years and life expectancy of ≥ 12 weeks. * Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1. * Resolution of (non-laboratory) adverse effects of recent surgery, radiotherapy, or chemotherapy to Grade ≤1 prior to first study treatment (with the exception of alopecia or neuropathy). * Patients must have had one prior platinum-based chemotherapeutic regimen for management of endometrial carcinoma or carcinosarcoma. Initial treatment may include chemotherapy, chemotherapy and radiation therapy, and/or consolidation/maintenance therapy. Chemotherapy administered in conjunction with primary radiation as a radiosensitizer WILL be counted as a systemic chemotherapy regimen. * Patients are allowed to have up to three prior cytotoxic regimens for management of recurrent or persistent disease. Hormonal therapies will not count toward the prior regimen limit. * Adequate normal organ and marrow function defined by the following laboratory results obtained within 14 days prior to first treatment: * Absolute neutrophil count (ANC) ≥ 1.5 x 10\^9/L (\> 1500 per mm\^3) * Platelet ≥ 75 x 10\^9/L (\>100,000 per mm\^3) * Serum bilirubin ≤ 1.5 x institutional upper limit of normal (ULN). (Unless Gilbert's Syndrome, for which Bilirubin ≤ 3 x institutional upper limit of normal (ULN), without concurrent clinically significant liver disease) * AST (SGOT)/ALT (SGPT) ≤ 3 x institutional upper limit of normal (ULN) unless liver metastases are present, in which case it must be ≤ 5x ULN * Serum creatinine ≤ 1.5 x institutional upper limit of normal (ULN) * Female subjects must either be of non-reproductive potential (i.e., post-menopausal by history: ≥60 years old and no menses for ≥1 year without an alternative medical cause and confirmed by FSH levels; OR history of hysterectomy, OR history of bilateral tubal ligation, OR history of bilateral oophorectomy) or must have a negative serum pregnancy test upon study entry. * Archival tissue, 15-20 unstained FFPE slides, must be available. If archival tissue is not available, patient will be required to undergo. * Subject is willing and able to comply with the protocol for the duration of the study including undergoing treatment, procedures (including on-treatment biopsy), and scheduled visits and examinations including follow up. * Patients must have been enrolled or agree to consent to the companion genomic profiling study MSKCC IRB# 12-245 - Patients must have signed an approved informed consent and authorization permitting release of personal information. * Women of childbearing potential (WOCBP) must agree to follow instructions for method(s) of contraception (See IDO Appendix 1) for the duration of treatment with study treatment(s) plus 5 months post-treatment completion. Exclusion Criteria: * Involvement in the planning and/or conduct of the study (applies to both Bristol-Myers Squibb staff and/or staff at the study site); * Known MMR-deficient patient and / or MSI-H will be excluded. Results of IMPACT #12-245 do not need to have resulted prior to enrollment. If results of IMPACT #12-245 shows MSI-H and / or MMR-deficient and patient has already started study, patient can continue on treatment * Prior enrollment in the present study or another clinical study with receipt of an investigational product during the last 4 weeks. * Any previous treatment with an IDO, PD-1 or PD-L1 inhibitor, or any anti-CTLA4. * History of another primary malignancy except for: * Malignancy treated with curative intent and with no known active disease ≥3 years before the first dose of study drug and of low potential risk for recurrence * Adequately treated non-melanoma skin cancer or lentigo maligna without evidence of disease * Adequately treated carcinoma in situ without evidence of disease (e.g., cervical cancer in situ) * Adequately treated stage 1 breast cancer. * Receipt of the last dose of anti-cancer therapy (chemotherapy, immunotherapy, targeted therapy, biologic therapy, tumor embolization, radiation therapy, monoclonal antibodies) \< 21 days prior to the first dose of study drug. Receipt of the last dose of hormonal therapy within \< 7 days prior to the first dose of study drug. * Less than 4 weeks since the patient underwent any major surgery (e.g., major: laparotomy, laparoscopy) There is no delay in treatment for minor procedures (e.g., central venous access catheter placement). * Mean QT interval corrected for heart rate (QTc) ≥470 ms calculated from 3 electrocardiograms (ECGs) using Fridericia formula * Current or prior use of immunosuppressive medication within 28 days before the first dose of BMS-986205 and Nivolumab, with the exceptions of intranasal and inhaled corticosteroids or systemic corticosteroids at physiological doses, which are not to exceed 10 mg/day of prednisone, or an equivalent corticosteroid. Patients who have received acute, low dose, systemic immunosuppressant medications (e.g., dexamethasone for nausea or steroids as CT scan contrast premedication) may be enrolled. * Any prior Grade ≥3 immune-related adverse event (irAE) while receiving any previous immunotherapy agent, or any unresolved irAE \> Grade 1 * Active or prior documented autoimmune disease within the past 2 years NOTE: Subjects with vitiligo, Grave's disease, or psoriasis not requiring systemic treatment (within the past 2 years) are not excluded. * Active or prior documented inflammatory bowel disease (e.g., Crohn's disease, ulcerative colitis) * History of primary immunodeficiency * History and/or confirmed pneumonitis or interstitial lung disease requiring steroids * History of allogeneic organ transplant * History of hypersensitivity to nivolumab * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, uncontrolled hypertension, unstable angina pectoris, cardiac arrhythmia, active peptic ulcer disease or gastritis, active bleeding diatheses including any subject known to have evidence of acute or chronic hepatitis B, hepatitis C or human immunodeficiency virus (HIV), or psychiatric illness/social situations that would limit compliance with study requirements or compromise the ability of the subject to give written informed consent * Known history of previous clinical diagnosis of tuberculosis * Symptomatic or uncontrolled brain metastases requiring concurrent treatment, inclusive of but not limited to surgery, radiation and/or corticosteroids. * History of leptomeningeal carcinomatosis * Uncontrolled seizures. * Receipt of live attenuated vaccination within 30 days prior to study entry or within 30 days of receiving BMS-986205 or nivolumab. * Any condition that, in the opinion of the investigator, would interfere with evaluation of study treatment or interpretation of patient safety or study results * Patients who are pregnant or breastfeeding or patients of reproductive potential who are not willing to employ effective birth control from screening to 5 months after the last dose of BMS-986205 and nivolumab combination therapy * Participants with conditions known to interfere significantly with the absorption of oral medication, as per investigator judgement. History of small or large bowel obstruction, perforation, bowel fistula or abscess, within 3 months of registration, including subjects with palliative gastric drainage catheters. Subjects with palliative diverting ileostomy or colostomy are allowed if they have been symptom free for more than 3 months. * Subjects with refractory ascites, defined as ascites needing drainage catheter or therapeutic paracentesis more often than every 4 weeks. * Participants with a personal or family (ie, in a first-degree relative) history or presence of cytochrome b5 reductase deficiency (previously called methemoglobin reductase deficiency) or other diseases that puts them at risk of methemoglobinemia. * Underlying G6PD deficiency, blood methemoglobin \> ULN, assessed in an arterial or venous blood sample or by co-oximetry. * History or presence of hypersensitivity or idiosyncratic reaction to methylene blue * Prior history of serotonin syndrome.